Turn signals not flashing

But when you replaced the LED
with old style bulbs you had the same issue.

The thing is the turning on of the headlights caused a change in the turn signal AND the E flasher so it must be something that have in common.

Perhaps as simple as a wire connector in the steering column turn signal switch in the wrong socket of the plastic connector housing.

I have had odd things happen with bad / missing grounds.

If you are missing a ground in a tail light for instance, the power will go in one filament then to their common grounding point, not finding a ground there, through the other filament and then back down that filaments wire to another bulbs filament that does have a good ground.

If all 3 filaments are the same wattage they will all light but be 1/3 as bright.

If one filament is 10 watts and the other filaments are 40 watts the 10 will light almost as normal and the other 2 may not look to light at all.

Electrons will find a path if one exists.